Abstract

We introduce the notion of primitive elements in arbitrary truncated p-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the “points of exact order N,” developed by Drinfeld and Katz–Mazur for elliptic curves.

1 Background

In this paper, we observe that Raynaud’s theory of Haar measures on finite flat group schemes [10] may be used to define a “non-triviality” condition on sections, which we call non-nullity. For groups of order p, we show that non-null sections are “generators” in the sense of Oort–Tate theory [11]. For truncated p-divisible groups, we use a non-nullity condition to define the notion of primitivity, generalizing the “points of exact order N” of Drinfeld [3] and Katz–Mazur [6].

In the case of elliptic curves, Drinfeld and Katz–Mazur go further and define full level structures. This allows them to construct and prove nice properties of integral models of modular curves at arbitrary levels in a very elegant fashion. We believe that our definition of primitive elements may be a first step toward defining full level structures in certain cases, as it was in previous work by one of us in the case \(\mu _p \times \mu _p\) [12]. However, for general p-divisible groups, we believe that new ideas are needed, and we hope that this work will lead to a better understanding of the issues involved in defining full level structures.

1.1 The problem of full level structures

To understand the problem of finding level structures, consider the following setup. Let S be a Noetherian scheme that is flat over \(\mathbb {Z}_{(p)}\), and let G be a finite flat group scheme such that \(G[1/p]:=G \times _S S[1/p]\) is étale-locally isomorphic to \((\mathbb {Z}/p^r\mathbb {Z})^g\) (for instance, S could be a Shimura variety classifying g / 2-dimensional abelian varieties with additional structure, and G could be the \(p^r\)-torsion of the universal abelian variety). A level structure on G is a map \((\mathbb {Z}/p^r\mathbb {Z})^g \rightarrow G\) that is like an isomorphism. The desired properties of level structures are best described scheme-theoretically. The set of full level structures \(\mathcal {F}_G\) should be a closed subscheme of \(\underline{\mathrm {Hom}}_S \bigl ( (\mathbb {Z}/p^r\mathbb {Z})^g,G \bigr )\) satisfying:

Since \(\underline{\mathrm {Hom}}_S \bigl ( (\mathbb {Z}/p^r\mathbb {Z})^g,G \bigr )\) is flat over \(\mathbb {Z}_{(p)}\), these conditions determine \(\mathcal {F}_G\) uniquely. However, in practice, it may be difficult to tell if a given homomorphism is full. For many purposes, \(\mathcal {F}_G\) is only useful if there is an explicit description of the ideal defining it.

1.2 Previous results

In the case where G embeds into a smooth curve C over S (for example if \(G=E[p^r]\) for an elliptic curve E), a satisfactory theory of full level structures has been built out of the ideas of Drinfeld [3]. However, Drinfeld’s definition crucially uses the fact that G is a Cartier divisor in C. Katz and Mazur developed a notion of “full set of sections,” which they show is equivalent to the Drinfeld level structure in the case that \(G \subset C\) [6, §1.10]. However, as Chai and Norman pointed out [1, Appendix], the Katz–Mazur definition does not give a flat space in general—it fails even for the relatively simple example of \(G=\mu _p \times \mu _p\).

More recently, one of the present authors developed a notion of full homomorphisms in the specific case \(G=\mu _p \times \mu _p\) [12].

1.3 Primitive elements

The first step in finding a basis for a free module is to find a primitive vector—that is, an element that can be extended to a basis. Analogously, a first step towards defining a notion of full level structure might be to define a notion of primitive element for group schemes. In addition, the notion of primitive element is needed to define the correct notion of “linear independence,” which is a key part of the method in [12] for \(G=\mu _p \times \mu _p\). In this paper we develop a formal theory of primitive elements, generalizing the ad hoc notion defined in [12].

1.4 Primitive elements and full homomorphisms

One may suggest defining a homomorphism \(\varphi :(\mathbb {Z}/p^r\mathbb {Z})^g \rightarrow G\) to be “full” if it sends primitive vectors to primitive vectors. Indeed, if G is constant, then this corresponds to the condition that the matrix of \(\varphi \) has linearly independent columns. However, the example of \(\mu _p \times \mu _p\) studied in [12] shows why this definition does not give a flat space of full homomorphisms. In that case, one may think of \(\varphi \) as a “\(2\times 2\)-matrix with coefficients in \(\mu _p\).” If \(\varphi \) sends primitive vectors to primitive vectors, then the columns are “linearly independent,” but the rows may not be—hence the elements cutting out the condition that the rows be “linearly independent” are p-torsion elements in the coordinate ring of the space of full homomorphisms. On the other hand, the main theorem of [12] implies that column conditions together with the row conditions give a flat space.

For a general group G, there is no obvious analog of the row conditions, so it is not clear how to generalize from primitive vectors to full homomorphisms. A new idea is needed.

1.5 Summary

Let S be a scheme, and let G be a finite locally free (commutative) group scheme over S. Let |G| denote the rank of G. We define a closed subscheme \(G^\times \subset G\), which we call the non-null subscheme. The ideal cutting out \(G^\times \) consists of invariant measures, as in Raynaud’s theory [10], on the Cartier dual of G. As a consequence of Raynaud’s results, \(G^\times \) is finite and locally free over S of rank \(|G|-1\). We think of \(G^\times \) as the group-scheme version of the set of non-zero elements of G.

There does not seem to be any completely satisfactory word to use here. Since the identity element in G(S) can perfectly well lie in \(G^\times (S)\) (as happens in the second example below when S is a scheme over \(\mathbb F_p\)), it would be extremely confusing say that elements in \(G^\times (S)\) are non-zero. Instead we have chosen to say that they are “non-null.”

As evidence that the notion of non-nullity is reasonable, we mention the following examples:

If \(G=\Gamma _S\), the constant group-scheme associated to a finite abelian group \(\Gamma \), then \(G^\times = (\Gamma \setminus \{0\})_S\), the scheme of non-identity sections.

If \(G=\mu _p\), then \(G^\times =\mu _p^\times \), the scheme of primitive roots of unity.

If G is an Oort–Tate group [11] (i.e. \(|G|=p\)), then \(G^\times \) coincides with the scheme of generators defined by Haines and Rapoport [5].

If G is a Raynaud group [10] (i.e. G has an action of \(\mathbb {F}_q\) and \(|G|=q\) for some power q of p), then \(G^\times \) coincides with the scheme of\(\mathbb {F}_q\)-generators defined by Katz–Mazur (c.f. [9])

We define primitive elements using non-nullity as follows. Assume that \(G=\mathcal {G}[p^r]\), the \(p^r\)-torsion subgroup of a p-divisible group \(\mathcal {G}\) of height h. Let \(G^{{{\mathrm{prim}}}} = G \times _{\mathcal {G}[p]} (\mathcal {G}[p])^\times \), where the map \(G \rightarrow \mathcal {G}[p]\) is given by multiplication by \(p^{r-1}\). It follows that the subscheme \(G^{{{\mathrm{prim}}}} \subset G\) is locally free over S of rank \((p^h-1)p^{h(r-1)}\).

In specific examples, we can identify \(G^{{{\mathrm{prim}}}}\):

If \(V=(\mathbb {Q}_p/\mathbb {Z}_p)^h\) and \(\mathcal {G}=V_S\) is a constant p-divisible group, then \(G^{{{\mathrm{prim}}}}\) is the scheme associated to the set of primitive vectors in the free \(\mathbb {Z}/p^r\mathbb {Z}\)-module \(V[p^r]\).

If \(G=\mu _{p^r}\), then \(G^{{{\mathrm{prim}}}}\) is the subscheme of primitive roots of unity.

If \(G=E[p^r]\) for an elliptic curve E, then \(G^{{{\mathrm{prim}}}}\) is the scheme of sections “of exact order \(p^r\)” defined by Drinfeld–Katz–Mazur [3, 6].

This justifies the notation \(G^{{{\mathrm{prim}}}}\)—it is meant to evoke both the notion of primitive vector in a free module, and primitive root of unity.

1.6 Applications to Shimura varieties

Let X be a Shimura variety over \(\mathbb {Q}\) that has a universal abelian variety A over it, and suppose \(\mathfrak {X}\) and \(\mathcal {A}\) are models for X and A that are flat over \(\mathbb {Z}_{(p)}\). Then, for each \(r>1\), there is an interesting cover \(X_1(p^r)\) of X given by adding the additional data of a point of order \(p^r\) in A.

The scheme \(\mathfrak {X}_1(p^r):=\mathcal {A}[p^r]^{{{\mathrm{prim}}}}\) is an integral model for \(X_1(p^r)\) that is finite and flat over \(\mathfrak {X}\). Since \(\mathfrak {X}\) is flat over \(\mathbb {Z}_{(p)}\), this implies that \(\mathfrak {X}_1(p^r)\) is the Zariski-closure of \(X_1(p)\) in \(\underline{\mathrm {Hom}}_\mathfrak {X}(\mathbb {Z}/p^r\mathbb {Z},\mathcal {A}[p^r])\). In particular, this “flat-closure” model, which is a priori only flat over \(\mathbb {Z}_{(p)}\), is actually flat over \(\mathfrak {X}\).

On the other hand, one can show that, except for modular curves (or the Drinfeld case), the scheme \(\mathfrak {X}_1(p^r)\) is not normal. In particular, \(\mathfrak {X}_1(p^r)\) is not the normalization of \(\mathfrak {X}\) in \(X_1(p^r)\), and this gives an example where the “normalization” and “flat closure” models differ.

This issue of non-normality makes us doubtful that these models will have direct application to the Langlands program. Instead, we view the theory of primitive elements as an interesting tool to use in the future study of integral models. For example, it would be interesting to consider combining the notion of primitive element with parahoric models of Shimura varieties, in analogy with the work of Pappas on Hilbert modular varieties [9]. Using the theory of Raynaud group schemes, Pappas produces a model for \(\Gamma _1(p)\)-type level that is normal (but not finite over the base).

2 Review of Raynaud’s Haar measures for finite flat group schemes

In this section we work over an affine base scheme \(S = {{\mathrm{Spec}}}(k)\), and G denotes a commutative group scheme over S that is finite, flat and finitely presented. So \(G = {{\mathrm{Spec}}}(A)\) with A locally free of finite rank as k-module. This rank is a locally constant function, denoted |G|, on S.

We write \(G'={{\mathrm{Spec}}}(A')\) for the Cartier dual of G; it is another object of the same kind as G, and \(|G'| = |G|\). Recall that A and \(A'\) are the k-duals of each other.

As Raynaud [10] points out, it is helpful to think about \(f \in A\) as a function on G and \(\mu \in A'\) as a measure on G, and then to write \(\langle \mu , f \rangle \in k\) for the natural pairing of \(\mu \) with f. Closely following Raynaud’s notation and conventions, we

write \(\star \) for the multiplication law on \(A'\) (intuitively, convolution of measures),

write 1 for the unit element in the ring A (intuitively, the constant function with value 1),

write \(\delta \) for the unit element in the ring \(A'\), i.e. the counit \(A \rightarrow k\) for the coalgebra A (intuitively, evaluation of functions at the identity element in the group G),

denote the natural A-module structure on \(A'\) by \(f \mu \) (intuitively, pointwise multiplication of a measure by a function), and

denote the natural \(A'\)-module structure on A by \(\mu \star f\) (intuitively, the convolution of a function by a measure).

A G-module is by definition a comodule for the coalgebra A, but, because A is locally free of finite rank as k-module, giving a G-module is the same as giving an \(A'\)-module M. For example, the \(A'\)-module structure on A reviewed above is the one corresponding to the natural G-module structure on A.

Given a G-module M, its submodule \(M^G\) of G-invariants consists of all elements in M annihilated by the augmentation ideal \(I'\) in \(A'\). For any k-algebra R there is a natural map

(We are abbreviating \(\otimes _k\) to \(\otimes \).) When (2.1) is an isomorphism for every k-algebra, one says that forming G-invariants in M commutes with extension of scalars. Bear in mind that M need not have this property, even when it is locally free of finite rank as k-module.

For the G-module A one has \(A^G = k\). So, in this example, it is evident that forming invariants does commute with extension of scalars. Now \(A'\) is of course an \(A'\)-module, i.e. a G-module, so we can form its submodule of G-invariants

Raynaud proves (in the discussion on page 277 of [10]) that \((A')^G\) is a direct summand of \(A'\), locally free of rank 1 as k-module. In other words, G-invariant measures on G form a line bundle over S. When \(D_G\) is free of rank 1 (not just locally so), a basis element \(\mu \) for it is called a Haar measure on G. A G-invariant measure \(\mu :A \rightarrow k\) is a Haar measure if and only if it is surjective.

a direct summand of A that is locally free of rank 1 as a k-module. Note that since \(A=I\oplus k\), where \(I=\ker (\delta )\) is the augmentation ideal, (2.2) implies that \(J_G\) is the annihilator of I in A. Raynaud also proves that

(1)

The natural pairing \(A' \otimes A \rightarrow k\) restricts to a perfect pairing between \(D_G\) and \(J_G\). So the line bundles \(D_G\) and \(J_G\) on S are canonically dual to each other.

It follows from (2) that, Zariski locally on S, the G-module \(A'\) is isomorphic to the G-module A. So forming G-invariants in \(A'\) (i.e. forming \(D_G\)) commutes with extension of scalars. (This useful fact is brought out by Moret-Bailly in the section of [8] in which he summarizes Raynaud’s work.)

2.1 Integration in stages

We need one more fact about Haar measures, namely an analog of the “integration in stages formula” in the theory of Haar measures on locally compact groups. It seems plausible that this is well-known, but since we do not know a reference, we will provide a proof.

Here \(H = {{\mathrm{Spec}}}(B)\), \(K={{\mathrm{Spec}}}(C)\) are objects of the same type as G, so B and C are locally free k-modules. Moreover A is faithfully flat over its subalgebra C, and B is the quotient of A by the ideal generated by the augmentation ideal \(I_C\) in C. Our goal is to understand invariant measures on G in terms of invariant measures on H and K. Dual to \(C \subset A\) and \(A \twoheadrightarrow B\) are the algebra homomorphisms

Proof

It is evident that \(\mu _G\) is independent of the choice of the lifting \(\tilde{\mu }_K\), because this lifting is well-defined modulo \((I_{B'}) A'\), and \(I_{B'}\) annihilates \(\mu _H\). The rest of the lemma is most easily understood in terms of integration in stages, as we will now see.

For any H-module M the invariant measure \(\mu _H\) gives rise to a k-linear map \(M \rightarrow M^H\), defined by \(m \mapsto \mu _H \star m\). (We use \(\star \) to denote the operation of an element in \(B'\) on an H-module.) Applying this to the H-module A, we obtain a k-linear map

$$\begin{aligned} \mathcal I_H : A \rightarrow A^H=C, \end{aligned}$$

given by convolution with \(\mu _H\) (intuitively, integration over the orbits of H on G). We claim that, if \(\mu _H\) is a Haar measure, then \(\mathcal I_H\) is surjective. Indeed, this is a special case of the following more general statement. Let \(T={{\mathrm{Spec}}}(D)\) be an affine S-scheme, and let \(X={{\mathrm{Spec}}}(E)\) be an H-torsor over T. Then the map \(\mathcal I_H : E \rightarrow D\) (given by convolution with the Haar measure \(\mu _H\)) is surjective. Surjectivity of \(\mathcal I_H\) is fpqc local, so we are reduced to the case in which \(X = H \times T\). Then \(\mathcal I_H\) is obviously surjective, because it is obtained by tensoring \(\mu _H : B \twoheadrightarrow k\) with D.

The map \(\mathcal I_H : A \rightarrow C\) is equivariant with respect to \(G \twoheadrightarrow K\) (and the natural actions of G on A and K on C), and the composition \(\mu :=\mu _K \circ \mathcal I_H : A \rightarrow k\) is G-equivariant, i.e. \(\mu \in D_G\). Unwinding the definitions, one sees that \(\mu = \mu _G\). The work we did shows that \(\mu _G\) is surjective when both \(\mu _H\), \(\mu _K\) are surjective, and hence that \(\mu _K \otimes \mu _H \mapsto \mu _G\) is an isomorphism from \(D_K \otimes D_H\) to \(D_G\). \(\square \)

3 Non-null elements in G

In this section we continue with k and G as in the previous section.

3.1 Definition of non-nullity of elements in G

The explicit description (2.2) of \(J_G\) shows that it is an ideal in A. We will refer to \(J_G\) as the non-nullity ideal. We denote by \(G^{\times } \hookrightarrow G\) the closed subscheme of G cut out by the ideal \(J_G\). Observe that \(G^{\times }={{\mathrm{Spec}}}(A/J_G)\) is locally free of rank \(|G|-1\) over S.

For every k-algebra R, \(G^{\times }(R)\) is a subset of G(R). We say that an element \(g \in G(R)\) is non-null when it lies in the subset \(G^{\times }(R)\). In the next subsections we will investigate this notion.

3.2 Non-nullity in the constant case

Start with a finite abelian group \(\Gamma \) and use it to build a constant group scheme G / S. Then A is the algebra of k-valued functions \(f:\Gamma \rightarrow k\), the ring structure being pointwise multiplication of functions. Then \(I_G=\{ f \in A : f(e_\Gamma ) =0 \}\) and \(J_G=\{ f \in A : f(\gamma ) =0 \quad \forall \, \gamma \ne e_\Gamma \}\). So \(A = J_G \oplus I_G\). In other words the scheme G decomposes as the disjoint union of two open (and closed) subschemes: \(G^{\times }\) and the identity section \(e_G(S)\). This example explains why we have chosen to call \(G^{\times }\) the closed subscheme of non-null elements in G.

3.3 Testing non-nullity using an overring \(R' \supset R\)

An R-valued point of G is given by a k-algebra homomorphism \(g:A \rightarrow R\). The element \(g \in G(R)\) is non-null if and only if the ring homomorphism \(g:A \rightarrow R\) is 0 on the ideal \(J_G\) in A. Consequently, if \(R \rightarrow R'\) is an injective k-algebra homomorphism, then \(g \in G(R)\) is non-null if and only if its image in \(G(R')\) is non-null.

If \(R'/R\) is faithfully flat, then \(R \rightarrow R'\) is injective. So the notion of non-nullity is fpqc local, and therefore continues to make sense for any base scheme (or even algebraic space) S.

3.4 Non-nullity in the étale case

Assume that G / S is étale. Then, locally in the étale topology, G is constant. It follows from the calculation in the previous subsection that A is the direct sum of the ideals \(J_G\) and \(I_G\). In other words, A is the cartesian product of the k-algebras \(A/J_G\) and \(A/I_G\). Therefore

the closed subscheme \(G^{\times }\) is also an open subscheme of G.

the identity section \(e_G : S \hookrightarrow G\) is an open and closed immersion, and

So, in the étale case, \(G^{\times }\) is again the open (and closed) subscheme of G obtained by deleting the image of the identity section \(S \rightarrow G\).

3.5 Behavior under base change

Consider a k-algebra R. For any scheme X / k we denote by \(X_R\) its base change to R. In particular we may base change G to R, obtaining a group scheme \(G_R={{\mathrm{Spec}}}(R \otimes A)\) over R. In our review of Haar measures, we mentioned that the natural map \(R \otimes J_G \hookrightarrow J_{G_R}\) is an isomorphism, which tells us that the natural morphism

is an isomorphism. In other words, forming \(G^{\times }\) from G commutes with extension of scalars.

3.6 Non-nullity for Oort–Tate groups

Now let us examine the notion of non-nullity in the case of Oort–Tate groups [11]. Our notion of non-nullity applies to all groups of order p over any base ring k, but in order to compare it to the notion of Oort–Tate generator we need to restrict attention to \(\Lambda \)-algebras, where \(\Lambda \) is the base ring considered in [11]. If \(\zeta \in \mathbb {Z}_p\) is a primitive \((p-1)\)-rst root of unity, then

Let k be a \(\Lambda \)-algebra (e.g., a \(\mathbb Z_p\)-algebra). Then, given suitable \(a,b \in k\), Oort–Tate construct a group \(G_{a,b}\) of order p over k, but we will fix a, b and just call the group G. The corresponding k-algebra is \(A = k[x]/(x^p-ax)\), and its augmentation ideal is generated by x. So the ideal \(J_G\) consists of all elements in A that are annihilated by x, and a short computation reveals that \(J_G\) is the k-submodule of A generated by \(x^{p-1}-a\). This shows that an element \(g \in G(k)\) is non-null in our sense if and only if g is a generator of G in the sense of Haines–Rapoport [5] (this notion of generator was first used by Deligne–Rapoport in [2, section V.2.6, p. 106]). Moreover, as Haines–Rapoport show [5, Remark 3.3.2], this is also equivalent to g having “exact order p” in the sense of Drinfeld–Katz–Mazur. This agreement suggests that the notion of non-nullity is a natural one.

Example 3.1

The above discussion applies to the group \(\mu _p\) of pth roots of unity. The result is that a section \(\zeta \in \mu _p(k)\) lies in \(\mu _p^{\times }\) if and only if \(\Phi _p(\zeta )=0\) (where \(\Phi _p(T) = 1 + T + \cdots + T^{p-1}\) is the cyclotomic polynomial). In other words, \(\mu _p^\times \) is the subscheme of primitive pth roots of unity.

3.7 Non-nullity for Raynaud groups

Raynaud groups are a natural generalization of Oort–Tate groups, and in this case, again, the notion of non-nullity agrees with a well-studied notion. We thank G. Pappas for communicating this generalization to us.

Let \(q=p^n\) be a power of p and let D be the ring defined analogously to \(\Lambda \), but with q in place of p (see [10, section 1.1]), and let k be a D-algebra. Given a suitable 2n-tuple \((\delta _1,\ldots ,\delta _n,\gamma _1,\ldots ,\gamma _n) \in k^{2n}\), Raynaud, in [10, Collolaire 1.5.1], defines a group scheme G over k with \(|G|=q\) together with an action of \(\mathbb {F}_q\) on G—that is, G is an \(\mathbb {F}_q\)-vector space scheme of dimension 1. The corresponding k-algebra is \(A=k[x_i]/(x_i^p-\delta _{i}x_{i+1})\) where i ranges over \(\{1,\ldots ,n\}\) and \(x_{n+1}:=x_1\). Then the augmentation ideal is generated by \((x_1,\ldots ,x_n)\), and using [4, Proposition 2.1], for example, one can see that \(J_{G}\) is the k-submodule of A generated by \((x_1\cdots x_n)^{p-1} - \delta _1 \cdots \delta _n\). By [9, Proposition 5.1.5], \(G^\times \) is the scheme of “\(\mathbb {F}_q\)-generators of G”, in the sense of Katz–Mazur.

3.8 Products

Consider groups \(G_1\), \(G_2\) over k. The corresponding k-algebras, augmentation ideals, and non-nullity ideals will be denoted \(A_i\), \(I_i\), \(J_i\) (for \(i = 1,2\)). The ring of regular functions for the group \(G=G_1 \times G_2\) is \(A = A_1 \otimes A_2\), and its augmentation ideal \(I_G\) is \((I_1 \otimes A_2) + (A_1 \otimes I_2)\). Therefore the ideal \(J_G\) in A annihilated by I is the intersection of the ideal annihilated by \(I_1\), namely \(J_1 \otimes A_2\), and the one annihilated by \(I_2\), namely \(A_1 \otimes J_2\). (It follows that \( J_G= J_1 \otimes J_2 , \) and so \(J_G\) is also the product of the ideals \(J_1 \otimes A_2\) and \(A_1 \otimes J_2\).)

In more geometrical language, we just verified that \(G^\times \) is the “union” of the closed subschemes \(G_1 \times G_2^\times \) and \(G_1^\times \times G_2\) of G (i.e. it is the smallest closed subscheme containing the two given closed subschemes).

Some care is required in this situation. Consider an R-valued point \(g=(g_1,g_2)\) of G. If \(g_1\) is non-null or \(g_2\) is non-null, then g is non-null. However, the converse is false, as is illustrated by the next example (when considering points with values in a ring that is not an integral domain).

Example 3.2

Let \((x,y) \in \mu _p \times \mu _p\). Then (x, y) is non-null if and only if \(\Phi _p(x)\Phi _p(y)\) vanishes. So, for this group, non-nullity coincides with the notion of primitivity introduced in [12].

3.9 Extensions

as in Sect. 2.1. We use the same system of notation: to G, H, K correspond k-algebras A, B, C respectively. Their augmentation ideals will be denoted \(I_A\), \(I_B\), \(I_C\), and their non-nullity ideals will be denoted \(J_A\), \(J_B\), \(J_C\). Applying Lemma 2.1 to the Cartier dual of G, we see that \(J_B \otimes J_C \simeq J_A\), just as in the special case when \(G=H \times K\). In fact Lemma 2.1 says more. It tells us that

$$\begin{aligned} J_A=(i^*)^{-1}(J_B) J_C, \end{aligned}$$

(3.1)

where \(i^*\) denotes the surjection \(A \twoheadrightarrow B=A/I_C A\) obtained from \(i: H \hookrightarrow G\). From this we obtain the following lemma.

Lemma 3.3

The closed subscheme \(G^{\times }\) of G contains both of the following closed subschemes of G:

commute. This arrow is unique, and it is a closed immersion. If K is étale over S, then the first item can be strengthened to the statement that the square (3.2) is cartesian.

Proof

We begin with the first item. The first item is true if and only if there exists an arrow j making the square (3.2) commute. This is the condition that \(i^*(J_A) \subset J_B\). That this condition holds follows from (3.1), which shows that \(i^*(J_A)\) is the product of the ideals \(J_B\) and \((i^*)(J_C)\) in B.

The first item can be strengthened to the statement that the square (3.2) is cartesian if and only if the inclusion \(i^*(J_A) \subset J_B\) is an equality. This is certainly the case when \(i^*(J_C)\) is the unit ideal in B.

When K is étale over S, we have seen that \(C = J_C \oplus I_C\). Therefore there exists \(f \in J_C\) such that \(1-f \in I_C\). The image of f under \(i^*\) is equal to 1, showing that \(i^*(J_C)\) is indeed the unit ideal in B.

Finally, the second item is true if and only if the ideal \(AJ_C\) contains the ideal \(J_A\). The truth of this is obvious from (3.1). \(\square \)

Remark 3.4

Let \(h \in H(R)\). The lemma implies that, if h is non-null for H, then it is non-null for G. It also implies that the converse is true provided that K is étale over S. In general the converse is false. For example, \((1,y) \in \mu _p(R) \times \mu _p(R)\) is non-null if and only if \(p\Phi _p(y)=0\), while \(y \in \mu _p(R)\) is non-null if and only if \(\Phi _p(y)=0\). These are equivalent conditions when p is invertible in R, but not in general.

4 Primitivity of points in truncated p-divisible groups

In this section we fix a prime number p.

4.1 Definition of primitivity

Now we consider a p-divisible group \(\mathcal {G}\) of height h over any base scheme S. For any positive integer i we are interested in the \(p^i\)-torsion \(\mathcal {G}[p^i]\) in \(\mathcal {G}\), but henceforth we abbreviate \(\mathcal {G}[p^i]\) to \(\mathcal {G}_i\). For any pair i, j of positive integers there is then a short exact (in the fppf sense) sequence

The arrow \(\mathcal {G}_{i+j}\twoheadrightarrow \mathcal {G}_j\) (strictly speaking, its composition with \(\mathcal {G}_j \hookrightarrow \mathcal {G}_{i+j}\)) is given by raising to the power \(p^i\), and it is finite locally free of rank \(p^{hi}\).

Let R be a k-algebra, let x be an R-valued point of \(\mathcal {G}_i\), and write \(\bar{x}\) for the image of x under the canonical homomorphism \(\mathcal {G}_i \twoheadrightarrow \mathcal {G}_1\) (raising to the power \(p^{i-1}\)). We say that x is primitive if \(\bar{x}\) is non-null in \(\mathcal {G}_1(R)\).

In other words, if we define \(\mathcal {G}_i^{{{\mathrm{prim}}}}\) as the fiber product making

(4.2)

cartesian, then \(x \in \mathcal {G}_i(R)\) is primitive if and only if it lies in the image of the R-points of \(\mathcal {G}_i^{{{\mathrm{prim}}}}\). Because the square is cartesian, we see that

\(\mathcal {G}_i^{{{\mathrm{prim}}}} \rightarrow \mathcal {G}_i\) is a closed immersion, and

Now \(\mathcal {G}_1^{\times }\) is finite, locally free of rank \(p^h-1\) over S, so we conclude that \(\mathcal {G}_i^{{{\mathrm{prim}}}}\) is finite, locally free of rank \((p^h-1)p^{h(i-1)}\) over S.

4.2 Comparison with points of exact order N on elliptic curves

In this subsection we fix i and put \(N = p^i\). Consider an elliptic curve E over S. Let \(E_N\) denote its N-torsion points. Then consider the following two closed subschemes of \(E_N\), namely

The closed subscheme \(E_N^{{{\mathrm{prim}}}}\) defined above, and

The closed subscheme, call it \(E_N^\sharp \), of points of exact order N in the sense of Drinfeld and Katz–Mazur (see [3, 6]).

We claim that (\(\star \)) \(E_N^{{{\mathrm{prim}}}}\) coincides with \(E_N^\sharp \).

We need to prove that (\(\star \)) holds for every elliptic curve E / S. We cannot see a priori a natural morphism between these two objects, so we proceed in the same way that similar problems are treated in [6].

Step 1
:

It is evident that (\(\star \)) holds when p is invertible on S, because \(E_N/S\) is then étale.

Step 2
:

Next we check that (\(\star \)) holds for E / S whenever S is flat over \(\mathbb Z\). In this situation \(E_N\), \(E_N^{{{\mathrm{prim}}}}\), and \(E_N^\sharp \) are flat over \(\mathbb Z\). Here we used that \(E_N/S\) is flat (standard), that \(E_N^{{{\mathrm{prim}}}}/S\) is flat (see Sect. 4.1), and that \(E_N^\sharp /S\) is flat (see [6, Thm. 5.1.1]). By Step 1 the closed subschemes \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) of \(E_N\) coincide over the locus in S where p is invertible, so the flatness of \(E_N\), \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) over \(\mathbb Z\) forces \(E_N^{{{\mathrm{prim}}}}\) to coincide with \(E_N^{\sharp }\) over all of S.

Step 3
:

Let \(\mathcal E\) denote the moduli stack (over \(\mathbb Z\)) of elliptic curves, and choose a presentation (see [7]) \(f : \mathcal M \twoheadrightarrow \mathcal E\) for it. Here f is étale and surjective, and \(\mathcal M\) is a smooth scheme of finite type over \(\mathbb Z\). Pulling back the universal elliptic curve on \(\mathcal E\), we obtain an elliptic curve \(\mathbf E\) on the scheme \(\mathcal M\). In the the terminology of [6], \(\mathbf E/\mathcal M\) is a “modular family.”

Now consider an elliptic curve E over an arbitrary base scheme S. We consider the product \(\mathcal M \times S\) and write \(p_1\), \(p_2\) for the two projections. We then have two elliptic curves over \(\mathcal M \times S\), namely \(p_1^*\mathbf E\) and \(p_2^*E\), and we form the \(\mathcal M \times S\)-scheme T of isomorphisms between \(p_1^*\mathbf E\) and \(p_2^*E\). Over T the elliptic curves \(\mathbf E\) and E become tautologically isomorphic; the resulting elliptic curve on T will be denoted \(\tilde{E}\). At this point we have a commutative diagram

(4.3)

in which both squares are cartesian. The two arrows in the bottom row exhibit T as the fiber product of \(\mathcal M\) and S over \(\mathcal E\), so \(T \rightarrow S\) is étale and surjective.

Now \(\mathcal M\) is flat over \(\mathbb Z\), so (\(\star \)) holds for \(\mathbf E/\mathcal M\). It follows that (\(\star \)) holds for \(\tilde{E}/T\). Here we used that the operations of forming \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) both commute with base change (use Sect. 3.5 and the first chapter of [6], especially their Corollary 1.3.7). So \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) become equal after the étale surjective base change \(T \rightarrow S\). By descent theory \(E_N^{{{\mathrm{prim}}}}\) and \(E_N^\sharp \) are themselves equal.

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Acknowledgements

PW was supported by the National Science Foundation under the Mathematical Sciences Postdoctoral Research Fellowship No. 1606255. We thank G. Boxer, B. Levin and K. Madapusi Pera for interesting conversations about integral models. We are grateful to T. Haines, G. Pappas, and M. Rapoport for helpful comments on a preliminary version of this paper. We thank the referees for comments and suggestions.

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